Sparse Random Geometric Graphs

• Sparse random geometric graphs are random graphs generated by embedding vertices in space where edges form only if points fall within a small distance, enforcing sparsity.
• They exhibit distinct regimes with fragmentation, motif structures, and localized spectral behavior, revealing how geometry shapes connectivity and eigenvector distributions.
• These models underpin rigorous methods in statistical inference, wireless network design, and motion planning, bridging theory with practical algorithmic applications.

Searching arXiv for recent and foundational work on sparse random geometric graphs to ground the article. Sparse random geometric graphs (RGGs) are random graphs whose adjacency structure is induced by an underlying geometric embedding, with sparsity enforced by a small connection radius, a bounded mean degree, or an edge density that vanishes relative to graph size, depending on the model. Across the literature, the term covers several non-equivalent regimes: shrinking-radius binomial graphs on bounded domains, fixed-radius supercritical Gilbert graphs on Poisson point processes, and high-dimensional spherical models with edge probability $p=\alpha/n$. In all of these settings, sparsity makes local geometry, fragmentation, and metric effects decisive, but it does so in model-specific ways rather than through a single universal definition (Aguilar-Sanchez et al., 2020, Lima et al., 2024, Liu et al., 2021).

1. Model families and notions of sparsity

The standard two-dimensional finite-window model studied in the spectral-topological literature is $G(n,\ell)$: $n$ vertices are sampled independently and uniformly on the unit square, and two vertices are joined whenever their Euclidean distance is at most $\ell\in[0,\sqrt{2}]$. In this model, $\ell=0$ yields only isolated vertices, $\ell=\sqrt{2}$ yields the complete graph, and the average degree is explicitly

$$k=(n-1)\left(\pi \ell^2-\frac{8}{3}\ell^3+\frac{1}{2}\ell^4\right),$$

so that for $\ell\ll 1$,

$k\approx (n-1)\pi \ell^2\sim n\pi \ell^2.$

This makes $n\ell^2$ the natural sparse-regime control parameter in two dimensions (Aguilar-Sanchez et al., 2020).

A closely related adjacency-spectrum model uses the unit square with periodic boundary conditions, i.e. the unit torus, and varies a radius $G(n,\ell)$0. There sparsity is described operationally by small $G(n,\ell)$1, low mean degree $G(n,\ell)$2, low probability of full connectivity, and fragmentation into many disconnected components. In the numerical study with $G(n,\ell)$3, the spacing-statistics regime $G(n,\ell)$4 spans a low-density fragmented phase through a connected phase toward the complete graph (Dettmann et al., 2016).

Another important notion of sparsity is the thermodynamic regime. For RGGs on the $G(n,\ell)$5-dimensional torus $G(n,\ell)$6, with adjacency rule $G(n,\ell)$7, the average degree is

$G(n,\ell)$8

The thermodynamic regime fixes $G(n,\ell)$9, equivalently $n$0. This is sparse in the graph-theoretic sense because $n$1 and degree does not diverge with $n$2 (Avrachenkov et al., 2019).

Sparse RGGs also arise in continuum percolation. In the Gilbert graph on a homogeneous Poisson point process $n$3, edges join points within a fixed radius $n$4, and the expected degree is $n$5. Here sparsity means fixed local connectivity scale rather than shrinking-radius finite-sample asymptotics. The regime of interest is supercritical, $n$6, so there exists a unique infinite connected component $n$7 on which large-scale metric questions can be posed (Lima, 2024, Lima et al., 2024).

High-dimensional sparse RGGs use yet another parameterization. In the spherical model $n$8, latent vectors are sampled uniformly on $n$9, and edges are created by an inner-product threshold chosen to give marginal edge probability $\ell\in[0,\sqrt{2}]$0. The sparse case of main interest is

$\ell\in[0,\sqrt{2}]$1

so average degree remains $\ell\in[0,\sqrt{2}]$2 while ambient dimension $\ell\in[0,\sqrt{2}]$3 varies (Liu et al., 2021). A related spectral study uses the same spherical threshold model $\ell\in[0,\sqrt{2}]$4 and investigates both $\ell\in[0,\sqrt{2}]$5 and the more general sparse regime $\ell\in[0,\sqrt{2}]$6 with $\ell\in[0,\sqrt{2}]$7 (Cao et al., 9 Jul 2025).

Model	Sparse regime in the cited work	Structural hallmark
$\ell\in[0,\sqrt{2}]$8 on the unit square	$\ell\in[0,\sqrt{2}]$9, $\ell=0$0	many isolated vertices and tiny components
Unit-torus adjacency RGG	small $\ell=0$1, $\ell=0$2	fragmented spectrum and motif effects
Thermodynamic torus RGG	$\ell=0$3	constant expected degree
One-dimensional circular RGG	$\ell=0$4 with $\ell=0$5	sparse but wedge/triangle counts still diverge
Spherical high-dimensional RGG	$\ell=0$6	sparse geometry versus Erdős–Rényi comparison

2. Fragmentation, motifs, and clustering

In low-dimensional finite-window RGGs, the first structural effect of sparsity is fragmentation. In the unit-square model, small $\ell=0$7 produces low expected degree, many isolated vertices, many tiny disconnected components, and a subcritical or pre-connectivity regime. The number of non-isolated vertices,

$\ell=0$8

is therefore a particularly informative observable: $\ell=0$9 at $\ell=\sqrt{2}$0, $\ell=\sqrt{2}$1 at $\ell=\sqrt{2}$2, and $\ell=\sqrt{2}$3 in sparse graphs. The fraction $\ell=\sqrt{2}$4 acts as an effective order parameter for departure from the fully isolated regime, and in the cited numerical study it almost completely controls the normalized Randić and harmonic indices as well as the averaged Shannon entropy of eigenvectors (Aguilar-Sanchez et al., 2020).

The torus adjacency-spectrum study makes the same fragmentation picture explicit from the spectral side: for small $\ell=\sqrt{2}$5, the graph is highly likely to contain many disconnected components, the spectrum is close to a superposition of independent component spectra, and localization is naturally interpreted as concentration of eigenstates on disconnected components (Dettmann et al., 2016). This is the sparse-RGG analogue of modular isolation.

A common extrapolation from non-geometric sparse random graph models does not hold uniformly for RGGs: sparsity need not force clustering to vanish. In the one-dimensional circular binomial RGG with adjacency

$\ell=\sqrt{2}$6

the sparse regime is defined by

$\ell=\sqrt{2}$7

Here expected degree is of order $\ell=\sqrt{2}$8, but wedge and triangle counts are both of order $\ell=\sqrt{2}$9, and the global clustering coefficient

$$k=(n-1)\left(\pi \ell^2-\frac{8}{3}\ell^3+\frac{1}{2}\ell^4\right),$$0

still satisfies

$$k=(n-1)\left(\pi \ell^2-\frac{8}{3}\ell^3+\frac{1}{2}\ell^4\right),$$1

in probability. Moreover,

$$k=(n-1)\left(\pi \ell^2-\frac{8}{3}\ell^3+\frac{1}{2}\ell^4\right),$$2

under the same sparse assumptions (Yuan et al., 19 Feb 2026). In this specific one-dimensional model, sparse geometry preserves transitivity rather than destroying it.

3. Spectral and eigenvector structure in sparse low-dimensional RGGs

Sparse RGGs display a characteristic spectral crossover from localized, near-Poisson behavior to delocalized, random-matrix behavior as connectivity increases. In the unit-square model $$k=(n-1)\left(\pi \ell^2-\frac{8}{3}\ell^3+\frac{1}{2}\ell^4\right),$$3, the spectral analysis is performed not on the binary adjacency matrix but on a randomly weighted symmetric matrix $$k=(n-1)\left(\pi \ell^2-\frac{8}{3}\ell^3+\frac{1}{2}\ell^4\right),$$4 whose off-diagonal entries are Gaussian on existing edges and whose diagonal entries are also random. This produces a Poisson ensemble (PE) limit at $$k=(n-1)\left(\pi \ell^2-\frac{8}{3}\ell^3+\frac{1}{2}\ell^4\right),$$5 and a GOE limit at $$k=(n-1)\left(\pi \ell^2-\frac{8}{3}\ell^3+\frac{1}{2}\ell^4\right),$$6. The spacing-ratio reference values are

$$k=(n-1)\left(\pi \ell^2-\frac{8}{3}\ell^3+\frac{1}{2}\ell^4\right),$$7

while eigenvector localization is measured through the participation ratio (“IPR” in the paper) and Shannon entropy $$k=(n-1)\left(\pi \ell^2-\frac{8}{3}\ell^3+\frac{1}{2}\ell^4\right),$$8 (Aguilar-Sanchez et al., 2020).

The same study finds that the topological sparse-to-dense crossover occurs at

$$k=(n-1)\left(\pi \ell^2-\frac{8}{3}\ell^3+\frac{1}{2}\ell^4\right),$$9

for $\ell\ll 1$0, $\ell\ll 1$1, and $\ell\ll 1$2. Thus the principal topological crossover scale is $\ell\ll 1$3, consistent with $\ell\ll 1$4. For spectral quantities, the fitted exponents differ: $\ell\ll 1$5 Most notably, the normalized average Shannon entropy is almost perfectly correlated with $\ell\ll 1$6, so average eigenvector delocalization can be inferred numerically from a simple count of non-isolated vertices (Aguilar-Sanchez et al., 2020).

For the plain $\ell\ll 1$7 adjacency spectrum on the two-dimensional torus, the low-radius regime is again Poisson-like, but now geometry generates an additional feature absent from many non-spatial models: sharp discrete peaks at integer eigenvalues caused by symmetric motifs, especially a large multiplicity at eigenvalue $\ell\ll 1$8. These peaks create a zero-spacing spike that must be removed before fitting the continuous part of the nearest-neighbor spacing distribution. Quantitatively, the Brody parameter rises from $\ell\ll 1$9 at $k\approx (n-1)\pi \ell^2\sim n\pi \ell^2.$0 to $k\approx (n-1)\pi \ell^2\sim n\pi \ell^2.$1 at $k\approx (n-1)\pi \ell^2\sim n\pi \ell^2.$2, with the main crossover around $k\approx (n-1)\pi \ell^2\sim n\pi \ell^2.$3–$k\approx (n-1)\pi \ell^2\sim n\pi \ell^2.$4 for $k\approx (n-1)\pi \ell^2\sim n\pi \ell^2.$5 (Dettmann et al., 2016).

Sparse Laplacian behavior is different but equally geometric. In the thermodynamic regime $k\approx (n-1)\pi \ell^2\sim n\pi \ell^2.$6, the regularized normalized Laplacian

$k\approx (n-1)\pi \ell^2\sim n\pi \ell^2.$7

is used to avoid singularities from zero degrees. For the $k\approx (n-1)\pi \ell^2\sim n\pi \ell^2.$8 model on the torus, the low-end spectrum obeys

$k\approx (n-1)\pi \ell^2\sim n\pi \ell^2.$9

and the paper concludes that the spectral dimension satisfies

$n\ell^2$0

in the thermodynamic regime (Avrachenkov et al., 2019). This indicates that sparse geometric randomness preserves the ambient diffusive dimension at low frequencies.

4. High-dimensional sparse RGGs and the disappearance of geometry

In high dimension, sparse RGGs can lose their geometric signature altogether. For the spherical model $n\ell^2$1 in the constant-sparsity regime $n\ell^2$2, the total variation distance to Erdős–Rényi satisfies

$n\ell^2$3

whenever

$n\ell^2$4

The same paper recalls a signed-triangle lower bound showing detectability when $n\ell^2$5, so the sparse indistinguishability threshold is pinned down up to polylogarithmic factors (Liu et al., 2021).

Global spectral statistics can wash out geometry even earlier. In the spherical threshold model $n\ell^2$6, if

$n\ell^2$7

then the empirical spectral distribution of

$n\ell^2$8

converges to the same limiting law $n\ell^2$9 as $G(n,\ell)$00. More generally, if $G(n,\ell)$01 and

$G(n,\ell)$02

then the empirical spectral distribution of

$G(n,\ell)$03

converges to the semicircle law (Cao et al., 9 Jul 2025). This suggests that global spectral observables can become Erdős–Rényi-like before the full graph law is completely indistinguishable.

An entropy-based high-dimensional analysis sharpens the dependence on geometry and connection rule. For hard RGGs on the torus, and for soft RGGs in both cube and torus geometries, the graph law converges to $G(n,\ell)$04 as $G(n,\ell)$05. By contrast, for hard RGGs in the cube with coordinate kurtosis greater than $G(n,\ell)$06, residual edge dependence survives, and the limiting maximum entropy is strictly below the Erdős–Rényi maximum. The same work derives an Edgeworth correction showing

$G(n,\ell)$07

for the Shannon entropy of the graph distribution (Baker et al., 14 Mar 2025). High-dimensional sparsity therefore does not by itself imply independent-edge behavior; the boundary geometry and the hard-versus-soft connection rule remain decisive.

5. Metric geometry, random walks, and algorithms

Sparse RGGs support a rich large-scale metric theory. In first-passage percolation (FPP) on the supercritical Gilbert graph built from a Poisson point process in $G(n,\ell)$08, with $G(n,\ell)$09 and i.i.d. nonnegative edge weights, the growth set $G(n,\ell)$10 satisfies a Euclidean-ball shape theorem: $G(n,\ell)$11 for all sufficiently large $G(n,\ell)$12, almost surely, under the stated moment assumptions. Under exponential-moment assumptions, the quantitative shape theorem improves this to

$G(n,\ell)$13

and passage times obey moderate deviations

$G(n,\ell)$14

for $G(n,\ell)$15. Geodesics remain within Hausdorff distance $G(n,\ell)$16 of the Euclidean segment with stretched-exponential tails (Lima, 2024, Lima et al., 2024).

Random-walk metrics retain geometric information in sparse supercritical two-dimensional torus RGGs. For effective resistance $G(n,\ell)$17, the proposed and numerically validated model is

$G(n,\ell)$18

with commute time given exactly by

$G(n,\ell)$19

The logarithmic term reflects the two-dimensional Laplacian Green’s function, the quadratic term the compact torus topology, and the quartic angular term the square anisotropy of the torus representation. This contrasts with the very dense regime, where commute times degenerate to an inverse-degree sum (Rupchin et al., 12 Jun 2026).

Algorithmically, sparse geometry can make exact distance problems easier than in general unit disk graphs. For two-dimensional RGGs with expected average degree $G(n,\ell)$20, $G(n,\ell)$21, the diameter can be computed asymptotically almost surely in

$G(n,\ell)$22

time on the square and

$G(n,\ell)$23

time on the torus. The square bound is minimized at $G(n,\ell)$24, yielding $G(n,\ell)$25. The same paper shows that iFUB achieves a speedup of $G(n,\ell)$26 over naive $G(n,\ell)$27 on square RGGs but still requires $G(n,\ell)$28 on torus RGGs, reflecting the algorithmic importance of boundary geometry (Bläsius et al., 17 Mar 2026).

6. Inference, transferability, and application-facing theory

Sparse RGGs have also become a setting for statistical inference from topology alone. For the toroidal $G(n,\ell)$29-RGG $G(n,\ell)$30, a test of the latent dimension $G(n,\ell)$31 is built from the contrast

$G(n,\ell)$32

Under

$G(n,\ell)$33

the statistic

$G(n,\ell)$34

whereas under $G(n,\ell)$35 it diverges at rate $G(n,\ell)$36 (Yuan et al., 13 Oct 2025). This is a sparse-but-not-ultra-sparse regime: edge density vanishes, but expected degree still diverges.

Learning-theoretic work uses sparse RGGs as wireless network models precisely because local interference yields bounded average degree rather than dense graphons. In the expanding-domain, fixed-density regime with connection radius $G(n,\ell)$37, the expected number of neighbors is

$G(n,\ell)$38

A transferability theory for GNNs is then obtained by comparing the RGG to a deterministic grid graph (DGG), assuming

$G(n,\ell)$39

and using integral-Lipschitz graph filters. Under these assumptions, performance differences between small and large sparse geometric graphs can be bounded explicitly (Camargo et al., 1 Oct 2025). A closely related theory for wireless conflict graphs induced by sparse RGGs assumes an operator-norm bound

$G(n,\ell)$40

between RGG and DGG conflict-graph adjacencies and derives

$G(n,\ell)$41

for an $G(n,\ell)$42-layer GNN (Camargo et al., 2 Jun 2026).

Sparse-RGG theory also transfers directly into sampling-based motion planning. Karaman and Frazzoli conjectured that a planner inherits a graph property if its underlying RGG has that property; the localization–tessellation framework proves this for several sparse roadmap models. In particular, PRM with

$G(n,\ell)$43

is probabilistically complete, and Bluetooth-PRM, Soft-PRM, and Embedded-PRM inherit analogous completeness guarantees under their respective sparse-graph conditions. Standard PRM is furthermore asymptotically $G(n,\ell)$44-optimal for some constant $G(n,\ell)$45, while Embedded-PRM is asymptotically optimal under

$G(n,\ell)$46

(Solovey et al., 2016).

Sparse RGGs are therefore not a single asymptotic object but a family of geometry-driven sparse graph models with distinct local, spectral, metric, and inferential behaviors. In low dimensions, sparsity foregrounds fragmentation, motif structure, and geometric random-walk effects; in high dimensions, it can render geometry statistically or spectrally invisible; and in applications, it supports both rigorous inference procedures and algorithmic transfer principles that depend essentially on locality rather than on dense-limit averaging.